Problem: Simplify the following expression and state the condition under which the simplification is valid. $k = \dfrac{x^2 - 81}{x + 9}$
Answer: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = x$ $ b = \sqrt{81} = 9$ So we can rewrite the expression as: $k = \dfrac{({x} + {9})({x} {-9})} {x + 9} $ We can divide the numerator and denominator by $(x + 9)$ on condition that $x \neq -9$ Therefore $k = x - 9; x \neq -9$